Theorem curf2cl 16871

Description: The curry functor at a morphism is a natural transformation. (Contributed by Mario Carneiro, 13-Jan-2017.)

Hypotheses

Ref	Expression
curf2.g	|- G = ( <. C , D >. curryF F )
curf2.a	|- A = ( Base ` C )
curf2.c	|- ( ph -> C e. Cat )
curf2.d	|- ( ph -> D e. Cat )
curf2.f	|- ( ph -> F e. ( ( C X.c D ) Func E ) )
curf2.b	|- B = ( Base ` D )
curf2.h	|- H = ( Hom ` C )
curf2.i	|- I = ( Id ` D )
curf2.x	|- ( ph -> X e. A )
curf2.y	|- ( ph -> Y e. A )
curf2.k	|- ( ph -> K e. ( X H Y ) )
curf2.l	|- L = ( ( X ( 2nd ` G ) Y ) ` K )
curf2.n	|- N = ( D Nat E )

Assertion

Ref	Expression
curf2cl	|- ( ph -> L e. ( ( ( 1st ` G ) ` X ) N ( ( 1st ` G ) ` Y ) ) )

Proof of Theorem curf2cl

Dummy variables z w f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		curf2.g	. . . 4 |- G = ( <. C , D >. curryF F )
2		curf2.a	. . . 4 |- A = ( Base ` C )
3		curf2.c	. . . 4 |- ( ph -> C e. Cat )
4		curf2.d	. . . 4 |- ( ph -> D e. Cat )
5		curf2.f	. . . 4 |- ( ph -> F e. ( ( C X.c D ) Func E ) )
6		curf2.b	. . . 4 |- B = ( Base ` D )
7		curf2.h	. . . 4 |- H = ( Hom ` C )
8		curf2.i	. . . 4 |- I = ( Id ` D )
9		curf2.x	. . . 4 |- ( ph -> X e. A )
10		curf2.y	. . . 4 |- ( ph -> Y e. A )
11		curf2.k	. . . 4 |- ( ph -> K e. ( X H Y ) )
12		curf2.l	. . . 4 |- L = ( ( X ( 2nd ` G ) Y ) ` K )
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	curf2 16869	. . 3 |- ( ph -> L = ( z e. B |-> ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) ) )
14		eqid 2622	. . . . . . . . . 10 |- ( C X.c D ) = ( C X.c D )
15	14, 2, 6	xpcbas 16818	. . . . . . . . 9 |- ( A X. B ) = ( Base ` ( C X.c D ) )
16		eqid 2622	. . . . . . . . 9 |- ( Hom ` ( C X.c D ) ) = ( Hom ` ( C X.c D ) )
17		eqid 2622	. . . . . . . . 9 |- ( Hom ` E ) = ( Hom ` E )
18		relfunc 16522	. . . . . . . . . . 11 |- Rel ( ( C X.c D ) Func E )
19		1st2ndbr 7217	. . . . . . . . . . 11 |- ( ( Rel ( ( C X.c D ) Func E ) /\ F e. ( ( C X.c D ) Func E ) ) -> ( 1st ` F ) ( ( C X.c D ) Func E ) ( 2nd ` F ) )
20	18, 5, 19	sylancr 695	. . . . . . . . . 10 |- ( ph -> ( 1st ` F ) ( ( C X.c D ) Func E ) ( 2nd ` F ) )
21	20	adantr 481	. . . . . . . . 9 |- ( ( ph /\ z e. B ) -> ( 1st ` F ) ( ( C X.c D ) Func E ) ( 2nd ` F ) )
22		opelxpi 5148	. . . . . . . . . 10 |- ( ( X e. A /\ z e. B ) -> <. X , z >. e. ( A X. B ) )
23	9, 22	sylan 488	. . . . . . . . 9 |- ( ( ph /\ z e. B ) -> <. X , z >. e. ( A X. B ) )
24		opelxpi 5148	. . . . . . . . . 10 |- ( ( Y e. A /\ z e. B ) -> <. Y , z >. e. ( A X. B ) )
25	10, 24	sylan 488	. . . . . . . . 9 |- ( ( ph /\ z e. B ) -> <. Y , z >. e. ( A X. B ) )
26	15, 16, 17, 21, 23, 25	funcf2 16528	. . . . . . . 8 |- ( ( ph /\ z e. B ) -> ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) : ( <. X , z >. ( Hom ` ( C X.c D ) ) <. Y , z >. ) --> ( ( ( 1st ` F ) ` <. X , z >. ) ( Hom ` E ) ( ( 1st ` F ) ` <. Y , z >. ) ) )
27		eqid 2622	. . . . . . . . . 10 |- ( Hom ` D ) = ( Hom ` D )
28	9	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ z e. B ) -> X e. A )
29		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ z e. B ) -> z e. B )
30	10	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ z e. B ) -> Y e. A )
31	14, 2, 6, 7, 27, 28, 29, 30, 29, 16	xpchom2 16826	. . . . . . . . 9 |- ( ( ph /\ z e. B ) -> ( <. X , z >. ( Hom ` ( C X.c D ) ) <. Y , z >. ) = ( ( X H Y ) X. ( z ( Hom ` D ) z ) ) )
32	31	feq2d 6031	. . . . . . . 8 |- ( ( ph /\ z e. B ) -> ( ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) : ( <. X , z >. ( Hom ` ( C X.c D ) ) <. Y , z >. ) --> ( ( ( 1st ` F ) ` <. X , z >. ) ( Hom ` E ) ( ( 1st ` F ) ` <. Y , z >. ) ) <-> ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) : ( ( X H Y ) X. ( z ( Hom ` D ) z ) ) --> ( ( ( 1st ` F ) ` <. X , z >. ) ( Hom ` E ) ( ( 1st ` F ) ` <. Y , z >. ) ) ) )
33	26, 32	mpbid 222	. . . . . . 7 |- ( ( ph /\ z e. B ) -> ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) : ( ( X H Y ) X. ( z ( Hom ` D ) z ) ) --> ( ( ( 1st ` F ) ` <. X , z >. ) ( Hom ` E ) ( ( 1st ` F ) ` <. Y , z >. ) ) )
34	11	adantr 481	. . . . . . 7 |- ( ( ph /\ z e. B ) -> K e. ( X H Y ) )
35	4	adantr 481	. . . . . . . 8 |- ( ( ph /\ z e. B ) -> D e. Cat )
36	6, 27, 8, 35, 29	catidcl 16343	. . . . . . 7 |- ( ( ph /\ z e. B ) -> ( I ` z ) e. ( z ( Hom ` D ) z ) )
37	33, 34, 36	fovrnd 6806	. . . . . 6 |- ( ( ph /\ z e. B ) -> ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) e. ( ( ( 1st ` F ) ` <. X , z >. ) ( Hom ` E ) ( ( 1st ` F ) ` <. Y , z >. ) ) )
38	3	adantr 481	. . . . . . . . 9 |- ( ( ph /\ z e. B ) -> C e. Cat )
39	5	adantr 481	. . . . . . . . 9 |- ( ( ph /\ z e. B ) -> F e. ( ( C X.c D ) Func E ) )
40		eqid 2622	. . . . . . . . 9 |- ( ( 1st ` G ) ` X ) = ( ( 1st ` G ) ` X )
41	1, 2, 38, 35, 39, 6, 28, 40, 29	curf11 16866	. . . . . . . 8 |- ( ( ph /\ z e. B ) -> ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) = ( X ( 1st ` F ) z ) )
42		df-ov 6653	. . . . . . . 8 |- ( X ( 1st ` F ) z ) = ( ( 1st ` F ) ` <. X , z >. )
43	41, 42	syl6eq 2672	. . . . . . 7 |- ( ( ph /\ z e. B ) -> ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) = ( ( 1st ` F ) ` <. X , z >. ) )
44		eqid 2622	. . . . . . . . 9 |- ( ( 1st ` G ) ` Y ) = ( ( 1st ` G ) ` Y )
45	1, 2, 38, 35, 39, 6, 30, 44, 29	curf11 16866	. . . . . . . 8 |- ( ( ph /\ z e. B ) -> ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) = ( Y ( 1st ` F ) z ) )
46		df-ov 6653	. . . . . . . 8 |- ( Y ( 1st ` F ) z ) = ( ( 1st ` F ) ` <. Y , z >. )
47	45, 46	syl6eq 2672	. . . . . . 7 |- ( ( ph /\ z e. B ) -> ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) = ( ( 1st ` F ) ` <. Y , z >. ) )
48	43, 47	oveq12d 6668	. . . . . 6 |- ( ( ph /\ z e. B ) -> ( ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) ( Hom ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) ) = ( ( ( 1st ` F ) ` <. X , z >. ) ( Hom ` E ) ( ( 1st ` F ) ` <. Y , z >. ) ) )
49	37, 48	eleqtrrd 2704	. . . . 5 |- ( ( ph /\ z e. B ) -> ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) e. ( ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) ( Hom ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) ) )
50	49	ralrimiva 2966	. . . 4 |- ( ph -> A. z e. B ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) e. ( ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) ( Hom ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) ) )
51		fvex 6201	. . . . . 6 |- ( Base ` D ) e. _V
52	6, 51	eqeltri 2697	. . . . 5 |- B e. _V
53		mptelixpg 7945	. . . . 5 |- ( B e. _V -> ( ( z e. B |-> ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) ) e. X_ z e. B ( ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) ( Hom ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) ) <-> A. z e. B ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) e. ( ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) ( Hom ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) ) ) )
54	52, 53	ax-mp 5	. . . 4 |- ( ( z e. B |-> ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) ) e. X_ z e. B ( ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) ( Hom ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) ) <-> A. z e. B ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) e. ( ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) ( Hom ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) ) )
55	50, 54	sylibr 224	. . 3 |- ( ph -> ( z e. B |-> ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) ) e. X_ z e. B ( ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) ( Hom ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) ) )
56	13, 55	eqeltrd 2701	. 2 |- ( ph -> L e. X_ z e. B ( ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) ( Hom ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) ) )
57		eqid 2622	. . . . . . . . . 10 |- ( Id ` C ) = ( Id ` C )
58	3	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> C e. Cat )
59	9	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> X e. A )
60		eqid 2622	. . . . . . . . . 10 |- (comp ` C ) = (comp ` C )
61	10	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> Y e. A )
62	11	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> K e. ( X H Y ) )
63	2, 7, 57, 58, 59, 60, 61, 62	catrid 16345	. . . . . . . . 9 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( K ( <. X , X >. (comp ` C ) Y ) ( ( Id ` C ) ` X ) ) = K )
64	2, 7, 57, 58, 59, 60, 61, 62	catlid 16344	. . . . . . . . 9 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( ( Id ` C ) ` Y ) ( <. X , Y >. (comp ` C ) Y ) K ) = K )
65	63, 64	eqtr4d 2659	. . . . . . . 8 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( K ( <. X , X >. (comp ` C ) Y ) ( ( Id ` C ) ` X ) ) = ( ( ( Id ` C ) ` Y ) ( <. X , Y >. (comp ` C ) Y ) K ) )
66	4	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> D e. Cat )
67		simpr1 1067	. . . . . . . . . 10 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> z e. B )
68		eqid 2622	. . . . . . . . . 10 |- (comp ` D ) = (comp ` D )
69		simpr2 1068	. . . . . . . . . 10 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> w e. B )
70		simpr3 1069	. . . . . . . . . 10 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> f e. ( z ( Hom ` D ) w ) )
71	6, 27, 8, 66, 67, 68, 69, 70	catlid 16344	. . . . . . . . 9 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( I ` w ) ( <. z , w >. (comp ` D ) w ) f ) = f )
72	6, 27, 8, 66, 67, 68, 69, 70	catrid 16345	. . . . . . . . 9 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( f ( <. z , z >. (comp ` D ) w ) ( I ` z ) ) = f )
73	71, 72	eqtr4d 2659	. . . . . . . 8 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( I ` w ) ( <. z , w >. (comp ` D ) w ) f ) = ( f ( <. z , z >. (comp ` D ) w ) ( I ` z ) ) )
74	65, 73	opeq12d 4410	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> <. ( K ( <. X , X >. (comp ` C ) Y ) ( ( Id ` C ) ` X ) ) , ( ( I ` w ) ( <. z , w >. (comp ` D ) w ) f ) >. = <. ( ( ( Id ` C ) ` Y ) ( <. X , Y >. (comp ` C ) Y ) K ) , ( f ( <. z , z >. (comp ` D ) w ) ( I ` z ) ) >. )
75		eqid 2622	. . . . . . . 8 |- (comp ` ( C X.c D ) ) = (comp ` ( C X.c D ) )
76	2, 7, 57, 58, 59	catidcl 16343	. . . . . . . 8 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( Id ` C ) ` X ) e. ( X H X ) )
77	6, 27, 8, 66, 69	catidcl 16343	. . . . . . . 8 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( I ` w ) e. ( w ( Hom ` D ) w ) )
78	14, 2, 6, 7, 27, 59, 67, 59, 69, 60, 68, 75, 61, 69, 76, 70, 62, 77	xpcco2 16827	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( <. K , ( I ` w ) >. ( <. <. X , z >. , <. X , w >. >. (comp ` ( C X.c D ) ) <. Y , w >. ) <. ( ( Id ` C ) ` X ) , f >. ) = <. ( K ( <. X , X >. (comp ` C ) Y ) ( ( Id ` C ) ` X ) ) , ( ( I ` w ) ( <. z , w >. (comp ` D ) w ) f ) >. )
79	36	3ad2antr1 1226	. . . . . . . 8 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( I ` z ) e. ( z ( Hom ` D ) z ) )
80	2, 7, 57, 58, 61	catidcl 16343	. . . . . . . 8 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( Id ` C ) ` Y ) e. ( Y H Y ) )
81	14, 2, 6, 7, 27, 59, 67, 61, 67, 60, 68, 75, 61, 69, 62, 79, 80, 70	xpcco2 16827	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( <. ( ( Id ` C ) ` Y ) , f >. ( <. <. X , z >. , <. Y , z >. >. (comp ` ( C X.c D ) ) <. Y , w >. ) <. K , ( I ` z ) >. ) = <. ( ( ( Id ` C ) ` Y ) ( <. X , Y >. (comp ` C ) Y ) K ) , ( f ( <. z , z >. (comp ` D ) w ) ( I ` z ) ) >. )
82	74, 78, 81	3eqtr4d 2666	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( <. K , ( I ` w ) >. ( <. <. X , z >. , <. X , w >. >. (comp ` ( C X.c D ) ) <. Y , w >. ) <. ( ( Id ` C ) ` X ) , f >. ) = ( <. ( ( Id ` C ) ` Y ) , f >. ( <. <. X , z >. , <. Y , z >. >. (comp ` ( C X.c D ) ) <. Y , w >. ) <. K , ( I ` z ) >. ) )
83	82	fveq2d 6195	. . . . 5 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( <. X , z >. ( 2nd ` F ) <. Y , w >. ) ` ( <. K , ( I ` w ) >. ( <. <. X , z >. , <. X , w >. >. (comp ` ( C X.c D ) ) <. Y , w >. ) <. ( ( Id ` C ) ` X ) , f >. ) ) = ( ( <. X , z >. ( 2nd ` F ) <. Y , w >. ) ` ( <. ( ( Id ` C ) ` Y ) , f >. ( <. <. X , z >. , <. Y , z >. >. (comp ` ( C X.c D ) ) <. Y , w >. ) <. K , ( I ` z ) >. ) ) )
84		eqid 2622	. . . . . 6 |- (comp ` E ) = (comp ` E )
85	20	adantr 481	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( 1st ` F ) ( ( C X.c D ) Func E ) ( 2nd ` F ) )
86	23	3ad2antr1 1226	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> <. X , z >. e. ( A X. B ) )
87		opelxpi 5148	. . . . . . 7 |- ( ( X e. A /\ w e. B ) -> <. X , w >. e. ( A X. B ) )
88	59, 69, 87	syl2anc 693	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> <. X , w >. e. ( A X. B ) )
89		opelxpi 5148	. . . . . . 7 |- ( ( Y e. A /\ w e. B ) -> <. Y , w >. e. ( A X. B ) )
90	61, 69, 89	syl2anc 693	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> <. Y , w >. e. ( A X. B ) )
91		opelxpi 5148	. . . . . . . 8 |- ( ( ( ( Id ` C ) ` X ) e. ( X H X ) /\ f e. ( z ( Hom ` D ) w ) ) -> <. ( ( Id ` C ) ` X ) , f >. e. ( ( X H X ) X. ( z ( Hom ` D ) w ) ) )
92	76, 70, 91	syl2anc 693	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> <. ( ( Id ` C ) ` X ) , f >. e. ( ( X H X ) X. ( z ( Hom ` D ) w ) ) )
93	14, 2, 6, 7, 27, 59, 67, 59, 69, 16	xpchom2 16826	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( <. X , z >. ( Hom ` ( C X.c D ) ) <. X , w >. ) = ( ( X H X ) X. ( z ( Hom ` D ) w ) ) )
94	92, 93	eleqtrrd 2704	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> <. ( ( Id ` C ) ` X ) , f >. e. ( <. X , z >. ( Hom ` ( C X.c D ) ) <. X , w >. ) )
95		opelxpi 5148	. . . . . . . 8 |- ( ( K e. ( X H Y ) /\ ( I ` w ) e. ( w ( Hom ` D ) w ) ) -> <. K , ( I ` w ) >. e. ( ( X H Y ) X. ( w ( Hom ` D ) w ) ) )
96	62, 77, 95	syl2anc 693	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> <. K , ( I ` w ) >. e. ( ( X H Y ) X. ( w ( Hom ` D ) w ) ) )
97	14, 2, 6, 7, 27, 59, 69, 61, 69, 16	xpchom2 16826	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( <. X , w >. ( Hom ` ( C X.c D ) ) <. Y , w >. ) = ( ( X H Y ) X. ( w ( Hom ` D ) w ) ) )
98	96, 97	eleqtrrd 2704	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> <. K , ( I ` w ) >. e. ( <. X , w >. ( Hom ` ( C X.c D ) ) <. Y , w >. ) )
99	15, 16, 75, 84, 85, 86, 88, 90, 94, 98	funcco 16531	. . . . 5 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( <. X , z >. ( 2nd ` F ) <. Y , w >. ) ` ( <. K , ( I ` w ) >. ( <. <. X , z >. , <. X , w >. >. (comp ` ( C X.c D ) ) <. Y , w >. ) <. ( ( Id ` C ) ` X ) , f >. ) ) = ( ( ( <. X , w >. ( 2nd ` F ) <. Y , w >. ) ` <. K , ( I ` w ) >. ) ( <. ( ( 1st ` F ) ` <. X , z >. ) , ( ( 1st ` F ) ` <. X , w >. ) >. (comp ` E ) ( ( 1st ` F ) ` <. Y , w >. ) ) ( ( <. X , z >. ( 2nd ` F ) <. X , w >. ) ` <. ( ( Id ` C ) ` X ) , f >. ) ) )
100	25	3ad2antr1 1226	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> <. Y , z >. e. ( A X. B ) )
101		opelxpi 5148	. . . . . . . 8 |- ( ( K e. ( X H Y ) /\ ( I ` z ) e. ( z ( Hom ` D ) z ) ) -> <. K , ( I ` z ) >. e. ( ( X H Y ) X. ( z ( Hom ` D ) z ) ) )
102	62, 79, 101	syl2anc 693	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> <. K , ( I ` z ) >. e. ( ( X H Y ) X. ( z ( Hom ` D ) z ) ) )
103	14, 2, 6, 7, 27, 59, 67, 61, 67, 16	xpchom2 16826	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( <. X , z >. ( Hom ` ( C X.c D ) ) <. Y , z >. ) = ( ( X H Y ) X. ( z ( Hom ` D ) z ) ) )
104	102, 103	eleqtrrd 2704	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> <. K , ( I ` z ) >. e. ( <. X , z >. ( Hom ` ( C X.c D ) ) <. Y , z >. ) )
105		opelxpi 5148	. . . . . . . 8 |- ( ( ( ( Id ` C ) ` Y ) e. ( Y H Y ) /\ f e. ( z ( Hom ` D ) w ) ) -> <. ( ( Id ` C ) ` Y ) , f >. e. ( ( Y H Y ) X. ( z ( Hom ` D ) w ) ) )
106	80, 70, 105	syl2anc 693	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> <. ( ( Id ` C ) ` Y ) , f >. e. ( ( Y H Y ) X. ( z ( Hom ` D ) w ) ) )
107	14, 2, 6, 7, 27, 61, 67, 61, 69, 16	xpchom2 16826	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( <. Y , z >. ( Hom ` ( C X.c D ) ) <. Y , w >. ) = ( ( Y H Y ) X. ( z ( Hom ` D ) w ) ) )
108	106, 107	eleqtrrd 2704	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> <. ( ( Id ` C ) ` Y ) , f >. e. ( <. Y , z >. ( Hom ` ( C X.c D ) ) <. Y , w >. ) )
109	15, 16, 75, 84, 85, 86, 100, 90, 104, 108	funcco 16531	. . . . 5 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( <. X , z >. ( 2nd ` F ) <. Y , w >. ) ` ( <. ( ( Id ` C ) ` Y ) , f >. ( <. <. X , z >. , <. Y , z >. >. (comp ` ( C X.c D ) ) <. Y , w >. ) <. K , ( I ` z ) >. ) ) = ( ( ( <. Y , z >. ( 2nd ` F ) <. Y , w >. ) ` <. ( ( Id ` C ) ` Y ) , f >. ) ( <. ( ( 1st ` F ) ` <. X , z >. ) , ( ( 1st ` F ) ` <. Y , z >. ) >. (comp ` E ) ( ( 1st ` F ) ` <. Y , w >. ) ) ( ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ` <. K , ( I ` z ) >. ) ) )
110	83, 99, 109	3eqtr3d 2664	. . . 4 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( ( <. X , w >. ( 2nd ` F ) <. Y , w >. ) ` <. K , ( I ` w ) >. ) ( <. ( ( 1st ` F ) ` <. X , z >. ) , ( ( 1st ` F ) ` <. X , w >. ) >. (comp ` E ) ( ( 1st ` F ) ` <. Y , w >. ) ) ( ( <. X , z >. ( 2nd ` F ) <. X , w >. ) ` <. ( ( Id ` C ) ` X ) , f >. ) ) = ( ( ( <. Y , z >. ( 2nd ` F ) <. Y , w >. ) ` <. ( ( Id ` C ) ` Y ) , f >. ) ( <. ( ( 1st ` F ) ` <. X , z >. ) , ( ( 1st ` F ) ` <. Y , z >. ) >. (comp ` E ) ( ( 1st ` F ) ` <. Y , w >. ) ) ( ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ` <. K , ( I ` z ) >. ) ) )
111	5	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> F e. ( ( C X.c D ) Func E ) )
112	1, 2, 58, 66, 111, 6, 59, 40, 67	curf11 16866	. . . . . . . 8 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) = ( X ( 1st ` F ) z ) )
113	112, 42	syl6eq 2672	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) = ( ( 1st ` F ) ` <. X , z >. ) )
114	1, 2, 58, 66, 111, 6, 59, 40, 69	curf11 16866	. . . . . . . 8 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` w ) = ( X ( 1st ` F ) w ) )
115		df-ov 6653	. . . . . . . 8 |- ( X ( 1st ` F ) w ) = ( ( 1st ` F ) ` <. X , w >. )
116	114, 115	syl6eq 2672	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` w ) = ( ( 1st ` F ) ` <. X , w >. ) )
117	113, 116	opeq12d 4410	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> <. ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) , ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` w ) >. = <. ( ( 1st ` F ) ` <. X , z >. ) , ( ( 1st ` F ) ` <. X , w >. ) >. )
118	1, 2, 58, 66, 111, 6, 61, 44, 69	curf11 16866	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` w ) = ( Y ( 1st ` F ) w ) )
119		df-ov 6653	. . . . . . 7 |- ( Y ( 1st ` F ) w ) = ( ( 1st ` F ) ` <. Y , w >. )
120	118, 119	syl6eq 2672	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` w ) = ( ( 1st ` F ) ` <. Y , w >. ) )
121	117, 120	oveq12d 6668	. . . . 5 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( <. ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) , ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` w ) >. (comp ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` w ) ) = ( <. ( ( 1st ` F ) ` <. X , z >. ) , ( ( 1st ` F ) ` <. X , w >. ) >. (comp ` E ) ( ( 1st ` F ) ` <. Y , w >. ) ) )
122	1, 2, 58, 66, 111, 6, 7, 8, 59, 61, 62, 12, 69	curf2val 16870	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( L ` w ) = ( K ( <. X , w >. ( 2nd ` F ) <. Y , w >. ) ( I ` w ) ) )
123		df-ov 6653	. . . . . 6 |- ( K ( <. X , w >. ( 2nd ` F ) <. Y , w >. ) ( I ` w ) ) = ( ( <. X , w >. ( 2nd ` F ) <. Y , w >. ) ` <. K , ( I ` w ) >. )
124	122, 123	syl6eq 2672	. . . . 5 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( L ` w ) = ( ( <. X , w >. ( 2nd ` F ) <. Y , w >. ) ` <. K , ( I ` w ) >. ) )
125	1, 2, 58, 66, 111, 6, 59, 40, 67, 27, 57, 69, 70	curf12 16867	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( z ( 2nd ` ( ( 1st ` G ) ` X ) ) w ) ` f ) = ( ( ( Id ` C ) ` X ) ( <. X , z >. ( 2nd ` F ) <. X , w >. ) f ) )
126		df-ov 6653	. . . . . 6 |- ( ( ( Id ` C ) ` X ) ( <. X , z >. ( 2nd ` F ) <. X , w >. ) f ) = ( ( <. X , z >. ( 2nd ` F ) <. X , w >. ) ` <. ( ( Id ` C ) ` X ) , f >. )
127	125, 126	syl6eq 2672	. . . . 5 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( z ( 2nd ` ( ( 1st ` G ) ` X ) ) w ) ` f ) = ( ( <. X , z >. ( 2nd ` F ) <. X , w >. ) ` <. ( ( Id ` C ) ` X ) , f >. ) )
128	121, 124, 127	oveq123d 6671	. . . 4 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( L ` w ) ( <. ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) , ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` w ) >. (comp ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` w ) ) ( ( z ( 2nd ` ( ( 1st ` G ) ` X ) ) w ) ` f ) ) = ( ( ( <. X , w >. ( 2nd ` F ) <. Y , w >. ) ` <. K , ( I ` w ) >. ) ( <. ( ( 1st ` F ) ` <. X , z >. ) , ( ( 1st ` F ) ` <. X , w >. ) >. (comp ` E ) ( ( 1st ` F ) ` <. Y , w >. ) ) ( ( <. X , z >. ( 2nd ` F ) <. X , w >. ) ` <. ( ( Id ` C ) ` X ) , f >. ) ) )
129	1, 2, 58, 66, 111, 6, 61, 44, 67	curf11 16866	. . . . . . . 8 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) = ( Y ( 1st ` F ) z ) )
130	129, 46	syl6eq 2672	. . . . . . 7 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) = ( ( 1st ` F ) ` <. Y , z >. ) )
131	113, 130	opeq12d 4410	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> <. ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) , ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) >. = <. ( ( 1st ` F ) ` <. X , z >. ) , ( ( 1st ` F ) ` <. Y , z >. ) >. )
132	131, 120	oveq12d 6668	. . . . 5 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( <. ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) , ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) >. (comp ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` w ) ) = ( <. ( ( 1st ` F ) ` <. X , z >. ) , ( ( 1st ` F ) ` <. Y , z >. ) >. (comp ` E ) ( ( 1st ` F ) ` <. Y , w >. ) ) )
133	1, 2, 58, 66, 111, 6, 61, 44, 67, 27, 57, 69, 70	curf12 16867	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( z ( 2nd ` ( ( 1st ` G ) ` Y ) ) w ) ` f ) = ( ( ( Id ` C ) ` Y ) ( <. Y , z >. ( 2nd ` F ) <. Y , w >. ) f ) )
134		df-ov 6653	. . . . . 6 |- ( ( ( Id ` C ) ` Y ) ( <. Y , z >. ( 2nd ` F ) <. Y , w >. ) f ) = ( ( <. Y , z >. ( 2nd ` F ) <. Y , w >. ) ` <. ( ( Id ` C ) ` Y ) , f >. )
135	133, 134	syl6eq 2672	. . . . 5 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( z ( 2nd ` ( ( 1st ` G ) ` Y ) ) w ) ` f ) = ( ( <. Y , z >. ( 2nd ` F ) <. Y , w >. ) ` <. ( ( Id ` C ) ` Y ) , f >. ) )
136	1, 2, 58, 66, 111, 6, 7, 8, 59, 61, 62, 12, 67	curf2val 16870	. . . . . 6 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( L ` z ) = ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) )
137		df-ov 6653	. . . . . 6 |- ( K ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ( I ` z ) ) = ( ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ` <. K , ( I ` z ) >. )
138	136, 137	syl6eq 2672	. . . . 5 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( L ` z ) = ( ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ` <. K , ( I ` z ) >. ) )
139	132, 135, 138	oveq123d 6671	. . . 4 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( ( z ( 2nd ` ( ( 1st ` G ) ` Y ) ) w ) ` f ) ( <. ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) , ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) >. (comp ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` w ) ) ( L ` z ) ) = ( ( ( <. Y , z >. ( 2nd ` F ) <. Y , w >. ) ` <. ( ( Id ` C ) ` Y ) , f >. ) ( <. ( ( 1st ` F ) ` <. X , z >. ) , ( ( 1st ` F ) ` <. Y , z >. ) >. (comp ` E ) ( ( 1st ` F ) ` <. Y , w >. ) ) ( ( <. X , z >. ( 2nd ` F ) <. Y , z >. ) ` <. K , ( I ` z ) >. ) ) )
140	110, 128, 139	3eqtr4d 2666	. . 3 |- ( ( ph /\ ( z e. B /\ w e. B /\ f e. ( z ( Hom ` D ) w ) ) ) -> ( ( L ` w ) ( <. ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) , ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` w ) >. (comp ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` w ) ) ( ( z ( 2nd ` ( ( 1st ` G ) ` X ) ) w ) ` f ) ) = ( ( ( z ( 2nd ` ( ( 1st ` G ) ` Y ) ) w ) ` f ) ( <. ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) , ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) >. (comp ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` w ) ) ( L ` z ) ) )
141	140	ralrimivvva 2972	. 2 |- ( ph -> A. z e. B A. w e. B A. f e. ( z ( Hom ` D ) w ) ( ( L ` w ) ( <. ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) , ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` w ) >. (comp ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` w ) ) ( ( z ( 2nd ` ( ( 1st ` G ) ` X ) ) w ) ` f ) ) = ( ( ( z ( 2nd ` ( ( 1st ` G ) ` Y ) ) w ) ` f ) ( <. ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) , ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) >. (comp ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` w ) ) ( L ` z ) ) )
142		curf2.n	. . 3 |- N = ( D Nat E )
143	1, 2, 3, 4, 5, 6, 9, 40	curf1cl 16868	. . 3 |- ( ph -> ( ( 1st ` G ) ` X ) e. ( D Func E ) )
144	1, 2, 3, 4, 5, 6, 10, 44	curf1cl 16868	. . 3 |- ( ph -> ( ( 1st ` G ) ` Y ) e. ( D Func E ) )
145	142, 6, 27, 17, 84, 143, 144	isnat2 16608	. 2 |- ( ph -> ( L e. ( ( ( 1st ` G ) ` X ) N ( ( 1st ` G ) ` Y ) ) <-> ( L e. X_ z e. B ( ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) ( Hom ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) ) /\ A. z e. B A. w e. B A. f e. ( z ( Hom ` D ) w ) ( ( L ` w ) ( <. ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) , ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` w ) >. (comp ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` w ) ) ( ( z ( 2nd ` ( ( 1st ` G ) ` X ) ) w ) ` f ) ) = ( ( ( z ( 2nd ` ( ( 1st ` G ) ` Y ) ) w ) ` f ) ( <. ( ( 1st ` ( ( 1st ` G ) ` X ) ) ` z ) , ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` z ) >. (comp ` E ) ( ( 1st ` ( ( 1st ` G ) ` Y ) ) ` w ) ) ( L ` z ) ) ) ) )
146	56, 141, 145	mpbir2and 957	1 |- ( ph -> L e. ( ( ( 1st ` G ) ` X ) N ( ( 1st ` G ) ` Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   <.cop 4183   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   X_cixp 7908   Basecbs 15857   Hom chom 15952  compcco 15953   Catccat 16325   Idccid 16326   Func cfunc 16514   Nat cnat 16601   X._c cxpc 16808   curry_F ccurf 16850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-hom 15966  df-cco 15967  df-cat 16329  df-cid 16330  df-func 16518  df-nat 16603  df-xpc 16812  df-curf 16854

This theorem is referenced by:  curfcl  16872